Question

Erythrocytes are red blood cells containing hemoglobin. In a saline solution they shrivel when the salt concentration is high and swell when the salt concentration is low. In a \(25^{\circ} \mathrm{C}\) aqueous solution of \(\mathrm{NaCl}\), whose freezing point is \(-0.406^{\circ} \mathrm{C}\), erythrocytes neither swell nor shrink. If we want to calculate the osmotic pressure of the solution inside the erythrocytes under these conditions, what do we need to assume? Why? Estimate how good (or poor) of an assumption this is. Make this assumption and calculate the osmotic pressure of the solution inside the erythrocytes.

Short answer

We need to assume that the NaCl solution inside the erythrocytes is an ideal solution. This assumption might not be completely accurate at higher concentrations of NaCl, but it simplifies our calculations and provides a reasonable estimation. Using the given freezing point depression and the van't Hoff equation, we can calculate the osmotic pressure of the solution inside the erythrocytes to be approximately 10.62 atm/kg water under the given conditions.

Step-by-step solution

Make an assumption

We will make the assumption that the NaCl solution inside the erythrocytes is an ideal solution. That means the solution obeys Raoult's law and colligative properties such as freezing point depression are directly proportional to the solute concentration.

Discuss the validity of the assumption

The assumption that the NaCl solution inside the erythrocytes is an ideal solution might not be completely accurate at higher concentrations of NaCl due to the presence of ions, which can result in deviations from an ideal behaviour. However, we can assume that at the given conditions, this assumption is reasonable, as it simplifies our calculations and provides a reasonable estimation of the osmotic pressure.

Calculate the molality of the NaCl solution

We are given the freezing point depression of the NaCl solution as \(\Delta T_f = 0.406^{\circ} \mathrm{C}\). We can use the freezing point depression equation to find the molality of NaCl: \(\Delta T_f = K_f\cdot m\) Where \(\Delta T_f\) is the freezing point depression, \(K_f\) is the cryoscopic constant for water (1.86 K·kg/mol at \(25^{\circ} \mathrm{C}\)), and \(m\) is the molality of the solution in mol NaCl/kg water. Now we can solve for the molality: \(m = \frac{\Delta T_f}{K_f} = \frac{0.406^{\circ} \mathrm{C}}{1.86 \frac{K \cdot kg}{mol}} = 0.2181 \frac{mol NaCl}{kg water}\)

Calculate the osmotic pressure

We can now calculate the osmotic pressure using the van't Hoff equation: \(\Pi = i \cdot m \cdot R \cdot T\) Where \(\Pi\) is the osmotic pressure, \(i\) is the van't Hoff factor (which is 2 for NaCl), \(m\) is the molality of the solution, \(R\) is the ideal gas constant (0.0821 L·atm/mol·K), and \(T\) is the temperature in Kelvin (25°C = 298 K). Plugging in the values, we get: \(\Pi = 2 \cdot 0.2181 \frac{mol NaCl}{kg water} \cdot 0.0821 \frac{L \cdot atm}{mol \cdot K} \cdot 298 K = 10.62 \frac{atm}{kg water}\) Therefore, the osmotic pressure of the solution inside the erythrocytes is approximately 10.62 atm/kg water under the given conditions.

Key concepts

Erythrocytes

Erythrocytes, commonly known as red blood cells, are essential components of our blood. They play a crucial role in transporting oxygen from the lungs to various body tissues and bringing carbon dioxide back to the lungs for exhalation.
Erythrocytes are remarkable for their flexibility, which allows them to travel through narrow blood vessels. They are filled with hemoglobin, the protein that gives blood its red color and is responsible for oxygen transport.
When erythrocytes are placed in solutions with different salt concentrations, they exhibit interesting behaviors. For example:

• In a hypertonic solution (high salt concentration), erythrocytes can shrivel, a process known as crenation.
• In a hypotonic solution (low salt concentration), they can swell and even burst, which is called hemolysis.

The balance of water and salt in erythrocytes is crucial for maintaining their shape and function.

Ideal Solutions

An ideal solution is a theoretical concept used in chemistry where the interactions between solute and solvent molecules are identical to those between each of the same type of molecules.
In ideal solutions, the properties are functions of the concentration of solute particles and not their nature. Raoult's Law is applied where the vapor pressure of an ideal solution is directly proportional to the mole fraction of the solvent. Such assumptions make calculations simpler and more predictable for understanding the behavior of solutions.
However, it's important to note that ideal solutions are rare. Most real solutions deviate from ideal behavior due to intermolecular forces that differ between solvent and solute. In the case of erythrocytes, the assumption that the solution inside is ideal helps simplify the calculation process.Though, this might not always reflect the real scenario due to high solute concentration and ion presence.

Colligative Properties

Colligative properties are those that depend on the number of solute particles in a solution, rather than the identity of those particles.
Examples include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.
These properties are crucial because they provide insights into concentrations and interactions in a solution. They help chemists determine molar masses and properties of solutes with minimal experimental data.
For instance, in the case of freezing point depression, the decrease in the freezing point is directly proportional to the molality of the solute. This is why we see erythrocytes neither shrinking nor expanding in isotonic solutions, where the solute concentration inside and outside the cell is balanced.

Freezing Point Depression

Freezing point depression is a phenomenon where the freezing point of a liquid (usually water) is lowered by adding another compound, such as salt.
This occurs because the added solute disrupts the formation of the solid lattice structure necessary for freezing, thus requiring a lower temperature to achieve the phase change.The magnitude of freezing point depression can be calculated using the formula:
\[\Delta T_f = K_f \cdot m\]
where \( \Delta T_f \) is the change in freezing point, \( K_f \) is the cryoscopic constant of the solvent, and \( m \) is the molality of the solute.In the context of erythrocytes, this property helps determine the molality of the salt solution in which the cells are suspended. Understanding freezing point depression also aids in understanding how erythrocytes are affected by their surrounding environment.

van't Hoff Equation

The van't Hoff equation is used to determine the osmotic pressure of a solution.
Osmotic pressure is the pressure required to prevent the flow of solvent into the solution via osmosis. It's crucial for maintaining cell equilibrium, such as in erythrocytes.The equation is given by:
\[\Pi = i \cdot m \cdot R \cdot T\]
where \( \Pi \) is the osmotic pressure, \( i \) is the van't Hoff factor, representing the number of particles the solute dissociates into (2 for NaCl), \( m \) is the molality of the solution, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.In real-world applications, this calculation helps predict how solutions will behave in biological systems, like the precise internal conditions of erythrocytes, crucial for their proper functioning. Understanding and applying the van't Hoff equation allows chemists and biologists to predict osmotic pressures in various scenarios.